I have had a mission like this before which ended up timing up on me. It is not that there is not enough time on this one. I don't know what to do to kind this contact or pirate once at the new destination.
Or you can circumvent that, go to the original target system and use the FSS to locate the mission USS. Or sometimes you get lucky and your target pirate will appear in the supercruise where you can either interdict him or tail him until he drops somewhere and then follow his wake.
Jump to Patosata and then hang around near the star until the informant appears on your contacts...
They will then provide you with a mission update to find the pirate, often they’ll send you back to the original target system and you’ll realise just how much FD enjoys wasting your time on something they believe is good content.
Just go back into SC and they will normally reappear. On rare occasions they decide to spawn near a secondary star and you might need to chase them.You can try jumping out and back into the system. That might re-trigger it.